A celebrated theorem of Mark Green from the 1980s describes the syzygies of a curve embedded by a line bundle of sufficiently large degree: if $L$ is a line bundle on a smooth projective curve $C$ of genus $g$ with $\deg(L)\geq 2g+1+p$, then $L$ satisfies property $N_p$; equivalently, the minimal free resolution of its homogeneous ideal is linear for the first $p$ steps. Since then, many efforts have been made to find analogues of Green’s theorem in broader settings. Two major directions are the study of linear syzygies of adjoint line bundles on higher-dimensional varieties, as predicted by syzygetic Fujita conjectures, and the study of higher-weight syzygies, encoded by properties $N_{d,p}$. In joint work with Wenbo Niu, we address both directions by studying the syzygies of tautological line bundle on symmetric products of curves. Let $C$ be a smooth projective curve of genus $g$, let $L$ be a line bundle on $C$, and let $T_{k+1,L}$ be the tautological line bundle on the symmetric product $C_{k+1}$, obtained by descent from $L^{\boxtimes (k+1)}$ on $C^{k+1}$. We prove that for each integer $d$ with $0\leq d\leq k$, if $\deg(L)\geq dg+2g+1+p$, then $T_{k+1,L}$ satisfies property $N_{k+2-d,p}$. This yields a family of sharp results on higher syzygies of higher weight for symmetric products of curves in arbitrary dimension, and in particular recovers Green’s classical theorem for curves. We also prove a sharp numerical criterion for the nefness of certain divisors on $C_{k+1}$. Taken together, these results provide strong evidence for syzygetic Fujita conjecture on property $N_p$ adjoint linear series and may be viewed as a natural higher-dimensional analogue of Green’s theorem.

Mirror Symmetry predicts that each Calabi-Yau variety X admits a mirror-dual Calabi-Yau variety Y, so that the symplectic Gromov-Witten (GW) invariants of X are computed from period integrals on Y. Mirror Symmetry inspired calculations of GW invariants have been achieved in several large families of cases yielding lots of data, yet we are still lacking a general approach. Comes in Intrinsic Mirror Symmetry (Gross-Siebert, 2022), which provides a general construction of Y from X. I will talk about an ongoing joint project with Siebert, where we show that the generating function of GW invariants of X equals the expansion of a canonical period integral on the Intrinsic Mirror Y of X. The project heavily relies on AI-assisted proofs, using the existing databasis of mirror-symmetric GW calculations.

We prove that the infinitesimal invariant of a higher Chow cycle of type (2,3-g) on a generic abelian variety of dimension g<4 gives rise to a meromorphic Siegel modular form of (virtual) weight (Sym^4, det^-1) with at most pole of order 1, and that this construction is functorial with respect to degeneration, namely the K-theory elevator for the cycle corresponds to the Siegel operator for the modular form.

For an orthogonal modular variety, we construct a complex which is defined in terms of lattices and elliptic modular forms, which resembles the Gersten complex in Milnor K-theory, and which has a morphism to the Gersten complex of the modular variety by the Borcherds lifting. This provides a formalism for approaching the higher Chow groups of the modular variety by special cycles and Borcherds products. The construction is an incorporation of the theory of Borcherds products and ideas from Milnor K-theory.